
Notice that the r.h.s. of (REF ) indicates the capacity region of MIMO Gaussian broadcast channel with private messages only, which has been derived in[1]}, [2]}. It means an upper bound of the optimal value of (REF ) can be achieved by setting the rate of common message zero. So the capacity in the case that \(\max \lbrace \mu _{1}, \mu _{2} \rbrace  \ge 1\)  is obtained by the capacity without common message. Throughout this section, we only focus on the case \(0 < \mu _{1}, \mu _{2} < 1\) .
